metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊13C4, C2.1D80, C16⋊3Dic5, C10.4D16, C40.11Q8, C10.2Q32, C20.12Q16, C4.1Dic20, C2.2Dic40, C22.8D40, C8.10Dic10, (C2×C80).5C2, (C2×C16).3D5, C5⋊3(C16⋊3C4), (C2×C4).71D20, (C2×C10).14D8, C20.54(C4⋊C4), C40⋊5C4.2C2, C40.110(C2×C4), (C2×C8).297D10, (C2×C20).370D4, C4.8(C4⋊Dic5), C8.14(C2×Dic5), C2.3(C40⋊5C4), C10.14(C2.D8), (C2×C40).370C22, SmallGroup(320,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C80⋊13C4
G = < a,b | a80=b4=1, bab-1=a-1 >
Smallest permutation representation of C80⋊13C4
►Regular action on 320 points
Generators in S320
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)
(1 317 138 206)(2 316 139 205)(3 315 140 204)(4 314 141 203)(5 313 142 202)(6 312 143 201)(7 311 144 200)(8 310 145 199)(9 309 146 198)(10 308 147 197)(11 307 148 196)(12 306 149 195)(13 305 150 194)(14 304 151 193)(15 303 152 192)(16 302 153 191)(17 301 154 190)(18 300 155 189)(19 299 156 188)(20 298 157 187)(21 297 158 186)(22 296 159 185)(23 295 160 184)(24 294 81 183)(25 293 82 182)(26 292 83 181)(27 291 84 180)(28 290 85 179)(29 289 86 178)(30 288 87 177)(31 287 88 176)(32 286 89 175)(33 285 90 174)(34 284 91 173)(35 283 92 172)(36 282 93 171)(37 281 94 170)(38 280 95 169)(39 279 96 168)(40 278 97 167)(41 277 98 166)(42 276 99 165)(43 275 100 164)(44 274 101 163)(45 273 102 162)(46 272 103 161)(47 271 104 240)(48 270 105 239)(49 269 106 238)(50 268 107 237)(51 267 108 236)(52 266 109 235)(53 265 110 234)(54 264 111 233)(55 263 112 232)(56 262 113 231)(57 261 114 230)(58 260 115 229)(59 259 116 228)(60 258 117 227)(61 257 118 226)(62 256 119 225)(63 255 120 224)(64 254 121 223)(65 253 122 222)(66 252 123 221)(67 251 124 220)(68 250 125 219)(69 249 126 218)(70 248 127 217)(71 247 128 216)(72 246 129 215)(73 245 130 214)(74 244 131 213)(75 243 132 212)(76 242 133 211)(77 241 134 210)(78 320 135 209)(79 319 136 208)(80 318 137 207)
G:=sub<Sym(320)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,317,138,206)(2,316,139,205)(3,315,140,204)(4,314,141,203)(5,313,142,202)(6,312,143,201)(7,311,144,200)(8,310,145,199)(9,309,146,198)(10,308,147,197)(11,307,148,196)(12,306,149,195)(13,305,150,194)(14,304,151,193)(15,303,152,192)(16,302,153,191)(17,301,154,190)(18,300,155,189)(19,299,156,188)(20,298,157,187)(21,297,158,186)(22,296,159,185)(23,295,160,184)(24,294,81,183)(25,293,82,182)(26,292,83,181)(27,291,84,180)(28,290,85,179)(29,289,86,178)(30,288,87,177)(31,287,88,176)(32,286,89,175)(33,285,90,174)(34,284,91,173)(35,283,92,172)(36,282,93,171)(37,281,94,170)(38,280,95,169)(39,279,96,168)(40,278,97,167)(41,277,98,166)(42,276,99,165)(43,275,100,164)(44,274,101,163)(45,273,102,162)(46,272,103,161)(47,271,104,240)(48,270,105,239)(49,269,106,238)(50,268,107,237)(51,267,108,236)(52,266,109,235)(53,265,110,234)(54,264,111,233)(55,263,112,232)(56,262,113,231)(57,261,114,230)(58,260,115,229)(59,259,116,228)(60,258,117,227)(61,257,118,226)(62,256,119,225)(63,255,120,224)(64,254,121,223)(65,253,122,222)(66,252,123,221)(67,251,124,220)(68,250,125,219)(69,249,126,218)(70,248,127,217)(71,247,128,216)(72,246,129,215)(73,245,130,214)(74,244,131,213)(75,243,132,212)(76,242,133,211)(77,241,134,210)(78,320,135,209)(79,319,136,208)(80,318,137,207)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,317,138,206)(2,316,139,205)(3,315,140,204)(4,314,141,203)(5,313,142,202)(6,312,143,201)(7,311,144,200)(8,310,145,199)(9,309,146,198)(10,308,147,197)(11,307,148,196)(12,306,149,195)(13,305,150,194)(14,304,151,193)(15,303,152,192)(16,302,153,191)(17,301,154,190)(18,300,155,189)(19,299,156,188)(20,298,157,187)(21,297,158,186)(22,296,159,185)(23,295,160,184)(24,294,81,183)(25,293,82,182)(26,292,83,181)(27,291,84,180)(28,290,85,179)(29,289,86,178)(30,288,87,177)(31,287,88,176)(32,286,89,175)(33,285,90,174)(34,284,91,173)(35,283,92,172)(36,282,93,171)(37,281,94,170)(38,280,95,169)(39,279,96,168)(40,278,97,167)(41,277,98,166)(42,276,99,165)(43,275,100,164)(44,274,101,163)(45,273,102,162)(46,272,103,161)(47,271,104,240)(48,270,105,239)(49,269,106,238)(50,268,107,237)(51,267,108,236)(52,266,109,235)(53,265,110,234)(54,264,111,233)(55,263,112,232)(56,262,113,231)(57,261,114,230)(58,260,115,229)(59,259,116,228)(60,258,117,227)(61,257,118,226)(62,256,119,225)(63,255,120,224)(64,254,121,223)(65,253,122,222)(66,252,123,221)(67,251,124,220)(68,250,125,219)(69,249,126,218)(70,248,127,217)(71,247,128,216)(72,246,129,215)(73,245,130,214)(74,244,131,213)(75,243,132,212)(76,242,133,211)(77,241,134,210)(78,320,135,209)(79,319,136,208)(80,318,137,207) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)], [(1,317,138,206),(2,316,139,205),(3,315,140,204),(4,314,141,203),(5,313,142,202),(6,312,143,201),(7,311,144,200),(8,310,145,199),(9,309,146,198),(10,308,147,197),(11,307,148,196),(12,306,149,195),(13,305,150,194),(14,304,151,193),(15,303,152,192),(16,302,153,191),(17,301,154,190),(18,300,155,189),(19,299,156,188),(20,298,157,187),(21,297,158,186),(22,296,159,185),(23,295,160,184),(24,294,81,183),(25,293,82,182),(26,292,83,181),(27,291,84,180),(28,290,85,179),(29,289,86,178),(30,288,87,177),(31,287,88,176),(32,286,89,175),(33,285,90,174),(34,284,91,173),(35,283,92,172),(36,282,93,171),(37,281,94,170),(38,280,95,169),(39,279,96,168),(40,278,97,167),(41,277,98,166),(42,276,99,165),(43,275,100,164),(44,274,101,163),(45,273,102,162),(46,272,103,161),(47,271,104,240),(48,270,105,239),(49,269,106,238),(50,268,107,237),(51,267,108,236),(52,266,109,235),(53,265,110,234),(54,264,111,233),(55,263,112,232),(56,262,113,231),(57,261,114,230),(58,260,115,229),(59,259,116,228),(60,258,117,227),(61,257,118,226),(62,256,119,225),(63,255,120,224),(64,254,121,223),(65,253,122,222),(66,252,123,221),(67,251,124,220),(68,250,125,219),(69,249,126,218),(70,248,127,217),(71,247,128,216),(72,246,129,215),(73,245,130,214),(74,244,131,213),(75,243,132,212),(76,242,133,211),(77,241,134,210),(78,320,135,209),(79,319,136,208),(80,318,137,207)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | + | - | + | + | - | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D5 | Q16 | D8 | Dic5 | D10 | D16 | Q32 | Dic10 | D20 | Dic20 | D40 | D80 | Dic40 |
| kernel | C80⋊13C4 | C40⋊5C4 | C2×C80 | C80 | C40 | C2×C20 | C2×C16 | C20 | C2×C10 | C16 | C2×C8 | C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of C80⋊13C4 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 14 | 98 |
| 0 | 143 | 192 |
| 64 | 0 | 0 |
| 0 | 44 | 3 |
| 0 | 78 | 197 |
G:=sub<GL(3,GF(241))| [1,0,0,0,14,143,0,98,192],[64,0,0,0,44,78,0,3,197] >;
C80⋊13C4 in GAP, Magma, Sage, TeX
C_{80}\rtimes_{13}C_4 % in TeX
G:=Group("C80:13C4"); // GroupNames label
G:=SmallGroup(320,62);
// by ID
G=gap.SmallGroup(320,62);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,675,192,1684,102,12550]);
// Polycyclic
G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
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